Question

$\lim_{x \to \0} \frac{3^{\frac{1}{x} } }{x}$=?
x->0 (la stanga si la dreapta)

Answer 1

Răspuns:

$\displaystyle\lim_{x\nearrow 0}\displaystyle\frac{3^{\frac{1}{x}}}{x}=\lim_{x\nearrow 0}\frac{1}{x}\cdot3^{\frac{1}{x}$

 Notăm $\frac{1}{x}=y\Rightarrow y\to-\infty$

Atunci limita devine

$\displaystyle\lim_{y\to-\infty}y\cdot 3^y=\lim_{y\to-\infty}\frac{y}{3^{-y}}=\lim_{y\to-\infty}\frac{1}{-3^{-y}\ln 3}=0$

(Am folosit l'Hospital)

$\displaystyle\lim_{x\searrow 0}\frac{3^{\frac{1}{x}}}{x}=\lim_{x\searrow 0}\frac{1}{x}\cdot 3^{\frac{1}{x}}=\infty\cdot\infty=\infty$

Explicație pas cu pas: